The flight is Zürich -> Dubai -> Sydney -> Adelaide.
Do I have to pick up my checked-in luggage in Sydney and check it in again before Adelaide?
Australia requires you to collect your baggage at your first point of entry into the country, which in your case would be Sydney.
Your bags may be tagged all the way through to Adelaide, but you will still need to collect them, take them through customs in Sydney, and then proceed to your domestic flight.
Presuming your domestic flight is with either Qantas or Virgin Australia then you will be able to drop your bags with them in the international terminal (follow the signs!) and then catch an airline-run bus to the domestic terminal. If you are flying with a different airline then you will need to catch the free T-Bus to the domestic terminal with your bags, and check in there. (You can also catch the train or a taxi, but these are not free like the T-bus)